Abstract
The first two sections of this paper deal with two parallelisms to the classical Bernstein and Jackson theory of best approximation by trigonometric polynomials. Starting from a generalization of the Bernstein inequality, the classes of functions with a logarithmic or exponential decay of the error of their best approximation are characterized by smoothness properties in terms of the moduli of continuity of suitable semi-group operators. The results may be understood as limiting cases of the classical situation where the best approximation error decays with some power of the parameter.
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